Warm Up
Compute the following by using long division.
(You have 10 minutes and no calculators.)
 1. 105/36
 2. 210/ 82
 3. 1956/ 15
 4. 12350/ 27
 5. 6239/ 18

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Slide 2.4 - 1
2.4
Real Zeros of
Polynomial
Functions
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Division Algorithm for Polynomials
Let f (x) and d(x) be polynomials with the degree of f
greater than or equal to the degree of d, and d(x)  0.
Then there are unique polynomials q(x) and r(x),
called the quotient and remainder, such that
f (x)  d(x)  q(x)  r(x)
where either r(x)  0 or the degree of r is less than the
degree of d.
The function f (x) in the division algorithm is the
dividend, and d(x) is the divisor.
If r(x)  0, we say d(x) divides evenly into f (x).
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Slide 2.4 - 3
Remainder Theorem
If polynomial f (x) is divided by x  k,
then the remainder is r  f (k).
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Slide 2.4 - 4
Example Using Polynomial Long Division
f(x) = x2 – 2x + 3 ÷ x – 1
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Examples
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Slide 2.4 - 6
Example Using the Remainder
Theorem
Find the remainder when f (x)  2x 2  x  12
is divided by x  3.
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Example


Use the Remainder Theorem to find the
remainder when f(x) is divided by x – k.
f(x) = x4 – 5; k = 1
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Example
Evaluate the function at x = 1
f(1) = (1)4 – 5
= -4
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Example Using Synthetic Division
Divide 3x 3  2x 2  x  5 by x 1 using synthetic division.
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Example Using Synthetic Division
Divide 3x 3  2x 2  x  5 by x 1 using synthetic division.
1 3
2
1
5
1 3
3
3
2
1
5
3
1
2
1
2
3
3x 3  2x 2  x  5
3
2
 3x  x  2 
x  1
x  1
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Slide 2.4 - 11
Rational Zeros Theorem
Suppose f is a polynomial function of degree n  1
of the form
f (x)  an x  an1 x
n
n1
 ...  a0 ,
with every coefficient an integer
and a0  0. If x  p / q is a rational zero of f ,
where p and q have no common integer factors
other than 1, then
g p is an integer factor of the constant coefficient a0 ,
g q is an integer factor of the leading coefficient an .
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Slide 2.4 - 12
Example Finding the Real Zeros of a
Polynomial Function
Find all of the real zeros of
f (x)  2x 5  x 4  2x 3  14x 2  6x  36
and identify them as rational or irrational.
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Slide 2.4 - 13
Example Finding the Real Zeros of a
Polynomial Function
Find all of the real zeros of
f (x)  2x 5  x 4  2x 3  14x 2  6x  36
and identify them as rational or irrational.
Potential Rational Zeros :
Factors of 36
: 1, 2, 3, 4, 6, 9, 12, 18, 36,
Factors of 2
1 3 9
 , ,
2 2 2
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Slide 2.4 - 14
Example Finding the Real Zeros of a
Polynomial Function
Find all of the real zeros of f (x)  2x 4  7x 3  8x 2 14x  8.
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Slide 2.4 - 15
Example Finding the Real Zeros of a
Polynomial Function
Find all of the real zeros of f (x)  2x 4  7x 3  8x 2 14x  8.
Potential Rational Zeros :
Factors of 8 1, 2, 4, 8
1
:
 1, 2, 4, 8, 
Factors of 2
1, 2
2
Compare the x-intercepts of the
graph and the list of possibilities,
and decide that 4 and  1/2 are
potential rational zeros.
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Slide 2.4 - 16
Example Finding the Real Zeros of a
Polynomial Function
Find all of the real zeros of f (x)  2x 4  7x 3  8x 2 14x  8.
4
2
2
7
8
8
1
14
8
4
 16
8
4
2
0
This tells us that
2x 4  7x 3  8x 2  14x  8  (x  4)(2x 3  x 2  4x  2).
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Slide 2.4 - 17
Example Finding the Real Zeros of a
Polynomial Function
Find all of the real zeros of f (x)  2x 4  7x 3  8x 2 14x  8.
1 / 2
4
2
1
0
2
0
4
0
2
2
1
This tells us that
1 2

2x  7x  8x  14 x  8  2(x  4)  x   x  2 .

2
4
3
2


1
The real zeros are 4,  ,  2.
2
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